[Paper Review] Boundary Conformal Field Theory and Entanglement Entropy in Two-Dimensional Quantum Lifshitz Critical Point

Oshikawa Masaki|arXiv (Cornell University)|Jul 21, 2010

Quantum many-body systems2 references19 citations

TL;DR

This paper resolves a discrepancy in the universal constant term of entanglement entropy for two-dimensional quantum Lifshitz critical points by re-examining the replica trick and boundary conformal field theory (CFT) approach. It identifies a flaw in the standard 'basis change' technique for compactified multi-component bosons, showing that new fields are not independently compactified. Using a geometric formulation based on compactification lattices, the authors derive the correct boundary entropy and confirm exact agreement with alternative calculations, establishing the universality of the constant term in the entanglement entropy for the free boson CFT.

ABSTRACT

I discuss the von Neumann entanglement entropy in two-dimensional quantum Lifshitz criical point, namely in Rokhsar-Kivelson type critical wavefunctions. I follow the approach proposed by B. Hsu et al. [Phys. Rev. B 79, 115421 (2009)], but point out a subtle problem concerning compactification of replica boson fields: although one can define a set of new boson fields by linear combinations of the original fields, the new fields are not compactified independently. In order to systematically study boundary conformal field theory of multicomponent free bosons, I employ a geometric formulation based on compactification lattices. The result from the boundary conformal field theory agrees exactly with alternative calculations by J.-M. Stephan et al. [Phys. Rev. B 80, 184421 (2009)], confirming its universality as argued originally by B. Hsu et al.

Motivation & Objective

• To resolve a discrepancy between two calculations of the universal constant term in the entanglement entropy for two-dimensional quantum Lifshitz critical points.
• To identify and correct a subtle flaw in the standard 'basis change' technique used in previous boundary CFT derivations of entanglement entropy.
• To systematically analyze boundary conformal field theory for multicomponent free bosons using a geometric formulation based on compactification lattices.
• To confirm the universality of the universal constant term in the entanglement entropy, as argued by Hsu et al., by deriving it consistently via lattice-based methods.

Proposed method

• Employ a geometric formulation of boundary CFT based on compactification lattices to describe multicomponent free boson systems.
• Use the multidimensional Poisson summation formula to relate partition functions in different bases, ensuring consistency with Cardy's consistency condition.
• Define boundary states (Dirichlet, Neumann, and mixed) via Ishibashi states and winding/zero-mode projections, with coefficients determined by consistency conditions.
• Derive the boundary entropy $ g_D $ and $ g_N $ for Dirichlet and Neumann boundary conditions using the volume of the compactification lattice $ v_0(ar{ heta}) $.
• Construct explicit boundary states using gluing conditions to ensure consistency with the Hamiltonian and modular invariance.
• Verify the derived amplitudes against known results, including the Dirichlet-Neumann amplitude $ z_{ND}( ilde{q}) = \frac{1}{\sqrt{2}} \tilde{q}^{-1/24} \prod_{n=1}^\infty \frac{1}{1 + \tilde{q}^n} $, which satisfies Cardy’s condition.

Experimental results

Research questions

• RQ1 Why does the standard basis change technique fail to correctly describe the entanglement entropy in multicomponent compactified boson CFTs?
• RQ2 What is the correct geometric formulation of boundary CFT for multicomponent free bosons with non-independent compactification?
• RQ3 How can the universal constant term in the entanglement entropy be consistently derived for the quantum Lifshitz universality class?
• RQ4 Does the boundary entropy $ g_D $ derived via lattice geometry agree with alternative calculations by Stéphan et al.?
• RQ5 Is the universal constant term in the entanglement entropy truly universal, as claimed by Hsu et al., when derived correctly?

Key findings

• The standard 'basis change' technique used in prior works is flawed because the new fields are not independently compactified, invalidating the derivation of the universal constant term.
• The correct boundary entropy for Dirichlet boundary conditions is $ g_D = (2g)^{-\mathcal{N}/4} (v_0(\Lambda))^{-1/2} $, derived from the compactification lattice geometry.
• The Neumann boundary entropy is $ g_N = \left(\frac{g}{2}\right)^{\mathcal{N}/4} (v_0(\Lambda))^{1/2} $, consistent with duality and lattice formulation.
• The Dirichlet-Neumann amplitude $ z_{ND} $ is independent of the compactification radius and matches the known expression involving Jacobi theta functions.
• The derived boundary entropy values satisfy Cardy’s consistency condition, confirming the validity of the geometric lattice approach.
• The final result for the universal constant term in the entanglement entropy exactly matches the alternative calculation by Stéphan et al., confirming its universality.